American Institute of Mathematical Sciences

Advanced
November  2016, 15(6): 2059-2074. doi: 10.3934/cpaa.2016027

On the Hardy-Littlewood-Sobolev type systems

Ze Cheng 1, , Genggeng Huang 2, and  Congming Li 3,

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado
2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
3. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
Keywords: existence, uniqueness., non-existence, Hardy-Littlewood-Sobolev.
Mathematics Subject Classification: Primary: 35J9.
Citation: Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141.  doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615.  doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2.  doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.   Google Scholar

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[13]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[14]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).   Google Scholar

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).   Google Scholar

[16]

E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.   Google Scholar

[18]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.   Google Scholar

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[20]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).   Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.   Google Scholar

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.   Google Scholar

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

show all references

References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141.  doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615.  doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2.  doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.   Google Scholar

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[13]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[14]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).   Google Scholar

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).   Google Scholar

[16]

E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.   Google Scholar

[18]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.   Google Scholar

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[20]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).   Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.   Google Scholar

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.   Google Scholar

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[1]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[2]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
[3]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[4]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[5]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[6]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[7]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[8]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205
[9]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[10]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[11]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[12]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[13]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
[14]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[15]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[16]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[17]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[18]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[19]

John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493
[20]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences